There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong — unsupported by research or evidence, or based on wishful logic.

Hacker overlooks the possibility that the mathematics learned in school, even if seldom applied directly, makes students better able to learn new quantitative skills. The on-the-job training in mathematics that Hacker envisions will go a whole lot better with an employee who gained a solid footing in math in school. […]

[In] teaching the specific skills that people need, you had better be confident that you’re going to cover all those skills. Because if you teach students the significance of the Consumer Price Index they are not going to know how to teach themselves the significance of projected inflation rates on their investment in CDs. Their practical knowledge will be specific to what you teach them, and won’t transfer.

The best bet for knowledge that can apply to new situations is an abstract understanding — seeing that apparently different problems have a similar underlying structure. And the best bet for students to gain this abstract understanding is to teach it explicitly.

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The thing is I know my times tables. Really know them and am proud to know them. And I mastered them by endless, thoughtless repetition when I was all of five years old.

Which is why it is all wrong that children should have compulsory maths until they are 18 as the Lords select committee on science and technology recommends.

This is addressing the problem of Britain’s maths-phobia from the wrong end. If you don’t know the maths you will need for most of life’s purposes – like avoiding being stiffed for 10 euros – by the time you are 16 then the education system has failed you big time.

There are two problems to do with maths education, it appears, and “experts” feel that these will be solved by making maths compulsory for everyone from ages 16 to 18 (report, 24 July).

One problem is that science and engineering students arrive at university without knowing the maths they will need for their disciplines.

The solution to this, obviously, is to teach more maths to science and engineering students, not to teach differential calculus to everyone else.

The other problem, we are told, is that many 16-year-olds still lack the level of numeracy required for citizenship in the modern world: they are “bewildered and bamboozled by numbers”.To manage our lives and to avoid being bamboozled by the many organisations and authorities who would like to pull numerical wool over our eyes, we non-scientists actually need quite a limited range of mathematics.

Basic arithmetic, a little geometry, the ability to read graphs, some elementary statistics and a little probability theory will probably suffice. All of these things can, in principle, be learnt by 16.

And if we have failed to teach them to young people in the first 11 years of their schooling, the solution is to look critically at what happens over that period, not to add a further two useless years of the same.

The call for “gold” open access (whether it comes from the report or the government) looks like yet another example of policy-makers failing to appreciate the differences in the structure of research across disciplines. In some areas (esp. the experimental sciences) research is only conducted by those with access to external funding, which would require only a slight increase to cover publishing costs. But in others (esp. “lone scholar” subjects such as mathematicsand the arts), a great deal of quality research is done by academic staff with no external funding, and also by retired staff (some with emeritus positions, some with no affiliation at all). Who is going to foot the bill for this research to be published in quality journals if we move to an “author-pays” model?

In my opinion, “mathematics education research” can be added to the list of disciplines disadvantaged by the “author pays” model of publishing.

Pedagogy and mathematics

It is undeniably important that mathematics teachers have mastered the topics they need to teach. The new Australian national curriculum is misguidedly increasing the amount of “statistics” of the school mathematics curriculum from less that 10% to as much as 40%. Many teachers are far from ready for the change.

But more often than not, the problem is not the mathematical expertise of the teachers. Pedagogical narrowness is a greater problem. Telling that there is a correct idea in a wrong solution to a problem on fractions requires unpacking of elementary concepts in a way that even an expert mathematician is not usually trained to do.

One of us – Jon – learned this only too well when he first taught future elementary school teachers their final university mathematics course.

Australian teachers at an elite private school could not understand one of Jon’s daughter’s Canadian long-division method nor her solution techniques for many advanced school topics. She got mediocre marks during the year because of this.

This year, the UK participated in the 53rd International Mathematical Olympiad, which was held this time in Mar del Plata, Argentina (about 500km south of Buenos Aires). The UK has been participating every year since 1967; it is now one of about a hundred countries which attend regularly.

The format is standard. Each country may send a team of up to six school-age students. However, all of them sit the competition on their own: it consists of two papers, each giving four-and-a-half hours for three questions. Medals are awarded according to rank order, conforming to the ratio gold:silver:bronze:no medal = 1:2:3:6.

Performance in the IMO depends on many factors. While we perform strongly compared to western European social democracies with similar or smaller populations, it has not been usual for the UK to outperform highly organised countries with larger populations than ours (such as Germany), or countries with elite state school systems that are able to provide intensive relevant education (such as Hungary). However, this year we did better than both Germany and Hungary.

In terms of the average scores achieved, this year was the hardest IMO for some years. It was gratifying to see our students trying hard, producing many ingenious ideas, and — often enough — succeeding in solving the problems where many others couldn’t. Indeed, all our students this year obtained a medal.

Of course, national one-upmanship is not the aim of the competition. One significant benefit is that it brings together like-minded students from all over the world. Indeed, the organisers did an excellent job of facilitating social interaction, providing a large room with all manner of games and diversions, and after the exams keeping it open (and supervised) almost permanently except for a few short hours around breakfast each day.

In my view, however, the best thing about the IMO is the quality of the mathematics: it exposes students to mathematical topics which are genuinely challenging yet approachable in a short timescale. Research inspired by IMO problems, both of a conventional and an experimental sort, has been on the rise in the last five years. Of this year’s problems, Q3 is an fascinating topic for further enquiry: is it possible to improve the bounds beyond those which are hinted at by the two parts of the problem, namely (2-ε)k and 2k?

The value extends beyond the six students who make it onto the UK’s team each year. More than a thousand students each year take part in our national olympiad, as an end in itself or with the hope of further participation.

It is also worth advertising the problems as a teaching resource. Recently, I have enjoyed using Q5 from IMO2010 as an introduction to Ackermann-style notations for large numbers: the problem provides a natural example of their use which students can enjoy experimenting with for themselves.

Those who are interested can read a blow-by-blow account of our participation in this year’s IMO. It remains to mention that, from an academic point of view, the UK’s olympiad activities are entirely volunteer-led. We are dependent on the goodwill and time of mathematicians all over the country: if you wish to join us, please send our organisation an email to let us know.

41. The Education Committee recommended that the Government should pilot a national syllabus in one large entry subject as part of the forthcoming A level reforms. We would recommend that maths should be the subject of such a pilot. […]

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